Antonyms for the phrase "shut door on" could include "welcome in," "invite," or "embrace." These words convey a sense of openness, acceptance, and hospitality, unlike shutting the door on someone or something which is often associated with rejection, exclusion, or dismissal. Other antonyms might include phrases like "make room for," "include," or "be receptive to," indicating a willingness to be open to new ideas, people, or experiences. Using such positive and welcoming language can help create an atmosphere of inclusivity and acceptance, fostering a sense of community and belonging for all.